Réorganisation de la librairie des réels

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@3311 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 202 insertions, 254 deletions

diff --git a/theories/Reals/Alembert.v b/theories/Reals/Alembert.v
index 0424b55f8..d3414317a 100644
--- a/theories/Reals/Alembert.v
+++ b/theories/Reals/Alembert.v
@@ -8,240 +8,12 @@
  
 (*i $Id$ i*)
 
-
-Require Max.
-Require Raxioms.
-Require DiscrR.
-Require Rbase.
+Require RealsB.
+Require Rfunctions.
 Require Rseries.
-Require Rtrigo_fun.
-
-(*********************)
-(* Lemmes techniques *)
-(*********************)
-
-Lemma not_sym : (r1,r2:R) ``r1<>r2`` -> ``r2<>r1``.
-Intros; Red; Intro H0; Rewrite H0 in H; Elim H; Reflexivity.
-Qed.
-
-Lemma Rgt_2_0 : ``0<2``.
-Sup0.
-Qed.
-
-Lemma Rgt_3_0 : ``0<3``.
-Sup0.
-Qed.
-
-Lemma tech1 : (An:nat->R;N:nat) ((n:nat)``(le n N)``->``0<(An n)``) -> ``0 < (sum_f_R0 An N)``.
-Intros; Induction N.
-Simpl; Apply H; Apply le_n.
-Simpl; Apply gt0_plus_gt0_is_gt0.
-Apply HrecN; Intros; Apply H; Apply le_S; Assumption.
-Apply H; Apply le_n.
-Qed.
-
-(* Relation de Chasles *)
-Lemma tech2 : (An:nat->R;m,n:nat) (lt m n) -> (sum_f_R0 An n) == (Rplus (sum_f_R0 An m) (sum_f_R0 [i:nat]``(An (plus (S m) i))`` (minus n (S m)))). 
-Intros; Induction n.
-Elim (lt_n_O ? H).
-Cut (lt m n)\/m=n.
-Intro; Elim H0; Intro.
-Replace (sum_f_R0 An (S n)) with ``(sum_f_R0 An n)+(An (S n))``; [Idtac | Reflexivity].
-Replace (minus (S n) (S m)) with (S (minus n (S m))).
-Replace (sum_f_R0 [i:nat](An (plus (S m) i)) (S (minus n (S m)))) with (Rplus (sum_f_R0 [i:nat](An (plus (S m) i)) (minus n (S m))) (An (plus (S m) (S (minus n (S m)))))); [Idtac | Reflexivity].
-Replace (plus (S m) (S (minus n (S m)))) with (S n).
-Rewrite (Hrecn H1).
-Ring.
-Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Do 2 Rewrite S_INR; Rewrite minus_INR.
-Rewrite S_INR; Ring.
-Apply lt_le_S; Assumption.
-Apply INR_eq; Rewrite S_INR; Repeat Rewrite minus_INR.
-Repeat Rewrite S_INR; Ring.
-Apply le_n_S; Apply lt_le_weak; Assumption.
-Apply lt_le_S; Assumption.
-Rewrite H1; Rewrite <- minus_n_n; Simpl.
-Replace (plus n O) with n; [Reflexivity | Ring].
-Inversion H.
-Right; Reflexivity.
-Left; Apply lt_le_trans with (S m); [Apply lt_n_Sn | Assumption].
-Qed.
-
-(* Somme d'une suite géométrique *)
-Lemma tech3 : (k:R;N:nat) ``k<>1`` -> (sum_f_R0 [i:nat](pow k i) N)==``(1-(pow k (S N)))/(1-k)``.
-Intros; Cut ``1-k<>0``.
-Intro; Induction N.
-Simpl; Rewrite Rmult_1r; Unfold Rdiv; Rewrite <- Rinv_r_sym.
-Reflexivity.
-Apply H0.
-Replace (sum_f_R0 ([i:nat](pow k i)) (S N)) with (Rplus (sum_f_R0 [i:nat](pow k i) N) (pow k (S N))); [Idtac | Reflexivity]; Rewrite HrecN; Replace ``(1-(pow k (S N)))/(1-k)+(pow k (S N))`` with ``((1-(pow k (S N)))+(1-k)*(pow k (S N)))/(1-k)``.
-Apply r_Rmult_mult with ``1-k``.
-Unfold Rdiv; Do 2 Rewrite <- (Rmult_sym ``/(1-k)``); Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [ Do 2 Rewrite Rmult_1l; Simpl; Ring | Apply H0].
-Apply H0.
-Unfold Rdiv; Rewrite Rmult_Rplus_distrl; Rewrite (Rmult_sym ``1-k``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym.
-Rewrite Rmult_1r; Reflexivity.
-Apply H0.
-Apply Rminus_eq_contra; Red; Intro; Elim H; Symmetry; Assumption.
-Qed.
-
-Lemma tech4 : (An:nat->R;k:R;N:nat) ``0<=k`` -> ((i:nat)``(An (S i))<k*(An i)``) -> ``(An N)<=(An O)*(pow k N)``.
-Intros; Induction N.
-Simpl; Right; Ring.
-Apply Rle_trans with ``k*(An N)``.
-Left; Apply (H0 N).
-Replace (S N) with (plus N (1)); [Idtac | Ring].
-Rewrite pow_add; Simpl; Rewrite Rmult_1r; Replace ``(An O)*((pow k N)*k)`` with ``k*((An O)*(pow k N))``; [Idtac | Ring]; Apply Rle_monotony.
-Assumption.
-Apply HrecN.
-Qed.
-
-Lemma tech5 : (An:nat->R;N:nat) (sum_f_R0 An (S N))==``(sum_f_R0 An N)+(An (S N))``.
-Intros; Reflexivity.
-Qed.
-
-Lemma tech6 : (An:nat->R;k:R;N:nat) ``0<=k`` -> ((i:nat)``(An (S i))<k*(An i)``) -> (Rle (sum_f_R0 An N) (Rmult (An O) (sum_f_R0 [i:nat](pow k i) N))).
-Intros; Induction N.
-Simpl; Right; Ring.
-Apply Rle_trans with (Rplus (Rmult (An O) (sum_f_R0 [i:nat](pow k i) N)) (An (S N))).
-Rewrite tech5; Do 2 Rewrite <- (Rplus_sym (An (S N))); Apply Rle_compatibility.
-Apply HrecN.
-Rewrite tech5 ; Rewrite Rmult_Rplus_distr; Apply Rle_compatibility.
-Apply tech4; Assumption.
-Qed.
-
-Lemma tech7 : (r1,r2:R) ``r1<>0`` -> ``r2<>0`` -> ``r1<>r2`` -> ``/r1<>/r2``.
-Intros; Red; Intro.
-Assert H3 := (Rmult_mult_r r1 ? ? H2).
-Rewrite <- Rinv_r_sym in H3; [Idtac | Assumption].
-Assert H4 := (Rmult_mult_r r2 ? ? H3).
-Rewrite Rmult_1r in H4; Rewrite <- Rmult_assoc in H4.
-Rewrite Rinv_r_simpl_m in H4; [Idtac | Assumption].
-Elim H1; Symmetry; Assumption.
-Qed.
-
-Lemma tech8 : (n,i:nat) (le n (plus (S n) i)).
-Intros; Induction i.
-Replace (plus (S n) O) with (S n); [Apply le_n_Sn | Ring].
-Replace (plus (S n) (S i)) with (S (plus (S n) i)).
-Apply le_S; Assumption.
-Apply INR_eq; Rewrite S_INR; Do 2 Rewrite plus_INR; Do 2 Rewrite S_INR; Ring.
-Qed.
-
-Lemma tech9 : (Un:nat->R) (Un_growing Un) -> ((m,n:nat)(le m n)->``(Un m)<=(Un n)``).
-Intros; Unfold Un_growing in H.
-Induction n.
-Induction m.
-Right; Reflexivity.
-Elim (le_Sn_O ? H0).
-Cut (le m n)\/m=(S n).
-Intro; Elim H1; Intro.
-Apply Rle_trans with (Un n).
-Apply Hrecn; Assumption.
-Apply H.
-Rewrite H2; Right; Reflexivity.
-Inversion H0.
-Right; Reflexivity.
-Left; Assumption.
-Qed.
-
-Lemma tech10 : (Un:nat->R;x:R) (Un_growing Un) -> (is_lub (EUn Un) x) -> (Un_cv Un x).
-Intros; Cut (bound (EUn Un)).
-Intro; Assert H2 := (Un_cv_crit ? H H1).
-Elim H2; Intros.
-Case (total_order_T x x0); Intro.
-Elim s; Intro.
-Cut (n:nat)``(Un n)<=x``.
-Intro; Unfold Un_cv in H3; Cut ``0<x0-x``.
-Intro; Elim (H3 ``x0-x`` H5); Intros.
-Cut (ge x1 x1).
-Intro; Assert H8 := (H6 x1 H7).
-Unfold R_dist in H8; Rewrite Rabsolu_left1 in H8.
-Rewrite Ropp_distr2 in H8; Unfold Rminus in H8.
-Assert H9 := (Rlt_anti_compatibility ``x0`` ? ? H8).
-Assert H10 := (Ropp_Rlt ? ? H9).
-Assert H11 := (H4 x1).
-Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H10 H11)).
-Apply Rle_minus; Apply Rle_trans with x.
-Apply H4.
-Left; Assumption.
-Unfold ge; Apply le_n.
-Apply Rgt_minus; Assumption.
-Intro; Unfold is_lub in H0; Unfold is_upper_bound in H0; Elim H0; Intros.
-Apply H4; Unfold EUn; Exists n; Reflexivity.
-Rewrite b; Assumption.
-Cut ((n:nat)``(Un n)<=x0``).
-Intro; Unfold is_lub in H0; Unfold is_upper_bound in H0; Elim H0; Intros.
-Cut (y:R)(EUn Un y)->``y<=x0``.
-Intro; Assert H8 := (H6 ? H7). 
-Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H8 r)).
-Unfold EUn; Intros; Elim H7; Intros.
-Rewrite H8; Apply H4.
-Intro; Case (total_order_Rle (Un n) x0); Intro.
-Assumption.
-Cut (n0:nat)(le n n0) -> ``x0<(Un n0)``. 
-Intro; Unfold Un_cv in H3; Cut ``0<(Un n)-x0``.
-Intro; Elim (H3 ``(Un n)-x0`` H5); Intros.
-Cut (ge (max n x1) x1).
-Intro; Assert H8 := (H6 (max n x1) H7).
-Unfold R_dist in H8.
-Rewrite Rabsolu_right in H8.
-Unfold Rminus in H8; Do 2 Rewrite <- (Rplus_sym ``-x0``) in H8.
-Assert H9 := (Rlt_anti_compatibility ? ? ? H8).
-Cut ``(Un n)<=(Un (max n x1))``.
-Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H10 H9)).
-Apply tech9; [Assumption | Apply le_max_l].
-Apply Rge_trans with ``(Un n)-x0``.
-Unfold Rminus; Apply Rle_sym1; Do 2 Rewrite <- (Rplus_sym ``-x0``); Apply Rle_compatibility.
-Apply tech9; [Assumption | Apply le_max_l].
-Left; Assumption.
-Unfold ge; Apply le_max_r.
-Apply Rlt_anti_compatibility with x0.
-Rewrite Rplus_Or; Unfold Rminus; Rewrite (Rplus_sym x0); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply H4; Apply le_n.
-Intros; Apply Rlt_le_trans with (Un n).
-Case (total_order_Rlt_Rle x0 (Un n)); Intro.
-Assumption.
-Elim n0; Assumption.
-Apply tech9; Assumption.
-Unfold bound; Exists x; Unfold is_lub in H0; Elim H0; Intros; Assumption.
-Qed.
-
-Lemma tech11 : (An,Bn,Cn:nat->R;N:nat) ((i:nat) (An i)==``(Bn i)-(Cn i)``) -> (sum_f_R0 An N)==``(sum_f_R0 Bn N)-(sum_f_R0 Cn N)``.
-Intros; Induction N.
-Simpl; Apply H.
-Do 3 Rewrite tech5; Rewrite HrecN; Rewrite (H (S N)); Ring.
-Qed.
-
-Lemma tech12 : (An:nat->R;x:R;l:R) (Un_cv [N:nat](sum_f_R0 [i:nat]``(An i)*(pow x i)`` N) l) -> (Pser An x l).
-Intros; Unfold Pser; Unfold infinit_sum; Unfold Un_cv in H; Assumption.
-Qed. 
-
-Lemma tech13 : (An:nat->R;k:R) ``0<=k<1`` -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) k) -> (EXT k0 : R | ``k<k0<1`` /\ (EX N:nat | (n:nat) (le N n)->``(Rabsolu ((An (S n))/(An n)))<k0``)).
-Intros; Exists ``k+(1-k)/2``.
-Split.
-Split.
-Pattern 1 k; Rewrite <- Rplus_Or; Apply Rlt_compatibility.
-Unfold Rdiv; Apply Rmult_lt_pos.
-Apply Rlt_anti_compatibility with k; Rewrite Rplus_Or; Replace ``k+(1-k)`` with R1; [Elim H; Intros; Assumption | Ring].
-Apply Rlt_Rinv; Apply Rgt_2_0.
-Apply Rlt_monotony_contra with ``2``.
-Apply Rgt_2_0.
-Unfold Rdiv; Rewrite Rmult_1r; Rewrite Rmult_Rplus_distr; Pattern 1 ``2``; Rewrite Rmult_sym; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]; Rewrite Rmult_1r; Replace ``2*k+(1-k)`` with ``1+k``; [Idtac | Ring].
-Elim H; Intros.
-Apply Rlt_compatibility; Assumption.
-Unfold Un_cv in H0; Cut ``0<(1-k)/2``.
-Intro; Elim (H0 ``(1-k)/2`` H1); Intros.
-Exists x; Intros.
-Assert H4 := (H2 n H3).
-Unfold R_dist in H4; Rewrite <- Rabsolu_Rabsolu; Replace ``(Rabsolu ((An (S n))/(An n)))`` with ``((Rabsolu ((An (S n))/(An n)))-k)+k``; [Idtac | Ring]; Apply Rle_lt_trans with ``(Rabsolu ((Rabsolu ((An (S n))/(An n)))-k))+(Rabsolu k)``.
-Apply Rabsolu_triang.
-Rewrite (Rabsolu_right k).
-Apply Rlt_anti_compatibility with ``-k``; Rewrite <- (Rplus_sym k); Repeat Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Repeat Rewrite Rplus_Ol; Apply H4.
-Apply Rle_sym1; Elim H; Intros; Assumption.
-Unfold Rdiv; Apply Rmult_lt_pos.
-Apply Rlt_anti_compatibility with k; Rewrite Rplus_Or; Elim H; Intros; Replace ``k+(1-k)`` with R1; [Assumption | Ring].
-Apply Rlt_Rinv; Apply Rgt_2_0.
-Qed.
-
-Definition SigT := Specif.sigT.
+Require SeqProp.
+Require PartSum.
+Require Max.
 
 (*************************************************)
 (* Différentes versions du critère de D'Alembert *)
@@ -252,7 +24,7 @@ Intros An H H0.
 Cut (sigTT R [l:R](is_lub (EUn [N:nat](sum_f_R0 An N)) l)) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)).
 Intro; Apply X.
 Apply complet.
-Unfold Un_cv in H0; Unfold bound; Cut ``0</2``; [Intro | Apply Rlt_Rinv; Apply Rgt_2_0].
+Unfold Un_cv in H0; Unfold bound; Cut ``0</2``; [Intro | Apply Rlt_Rinv; Sup0].
 Elim (H0 ``/2`` H1); Intros.
 Exists ``(sum_f_R0 An x)+2*(An (S x))``.
 Unfold is_upper_bound; Intros; Unfold EUn in H3; Elim H3; Intros.
@@ -263,10 +35,10 @@ Replace (sum_f_R0 An x) with (Rplus (sum_f_R0 An x1) (sum_f_R0 [i:nat](An (plus
 Pattern 1 (sum_f_R0 An x1); Rewrite <- Rplus_Or; Rewrite Rplus_assoc; Apply Rle_compatibility.
 Left; Apply gt0_plus_gt0_is_gt0.
 Apply tech1; Intros; Apply H.
-Apply Rmult_lt_pos; [Apply Rgt_2_0 | Apply H].
+Apply Rmult_lt_pos; [Sup0 | Apply H].
 Symmetry; Apply tech2; Assumption.
 Rewrite b; Pattern 1 (sum_f_R0 An x); Rewrite <- Rplus_Or; Apply Rle_compatibility.
-Left; Apply Rmult_lt_pos; [Apply Rgt_2_0 | Apply H].
+Left; Apply Rmult_lt_pos; [Sup0 | Apply H].
 Replace (sum_f_R0 An x1) with (Rplus (sum_f_R0 An x) (sum_f_R0 [i:nat](An (plus (S x) i)) (minus x1 (S x)))).
 Apply Rle_compatibility.
 Cut (Rle (sum_f_R0 [i:nat](An (plus (S x) i)) (minus x1 (S x))) (Rmult (An (S x)) (sum_f_R0 [i:nat](pow ``/2`` i) (minus x1 (S x))))).
@@ -278,11 +50,11 @@ Rewrite tech3.
 Replace ``1-/2`` with ``/2``.
 Unfold Rdiv; Rewrite Rinv_Rinv.
 Pattern 3 ``2``; Rewrite <- Rmult_1r; Rewrite <- (Rmult_sym ``2``); Apply Rle_monotony.
-Left; Apply Rgt_2_0.
+Left; Sup0.
 Left; Apply Rlt_anti_compatibility with ``(pow (/2) (S (minus x1 (S x))))``.
 Replace ``(pow (/2) (S (minus x1 (S x))))+(1-(pow (/2) (S (minus x1 (S x)))))`` with R1; [Idtac | Ring].
 Rewrite <- (Rplus_sym ``1``); Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rlt_compatibility.
-Apply pow_lt; Apply Rlt_Rinv; Apply Rgt_2_0.
+Apply pow_lt; Apply Rlt_Rinv; Sup0.
 DiscrR.
 Apply r_Rmult_mult with ``2``.
 Rewrite Rminus_distr; Rewrite <- Rinv_r_sym.
@@ -292,7 +64,7 @@ DiscrR.
 Pattern 3 R1; Replace R1 with ``/1``; [Apply tech7; DiscrR | Apply Rinv_R1].
 Replace (An (S x)) with (An (plus (S x) O)).
 Apply (tech6 [i:nat](An (plus (S x) i)) ``/2``).
-Left; Apply Rlt_Rinv; Apply Rgt_2_0.
+Left; Apply Rlt_Rinv; Sup0.
 Intro; Cut (n:nat)(ge n x)->``(An (S n))</2*(An n)``.
 Intro; Replace (plus (S x) (S i)) with (S (plus (S x) i)).
 Apply H6; Unfold ge; Apply tech8.
@@ -342,7 +114,7 @@ Rewrite Rminus_distr; Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym.
 Ring.
 DiscrR.
 DiscrR.
-Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rgt_2_0].
+Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0].
 Cut (n:nat)``/2*(Rabsolu (An n))<=(Wn n)<=(3*/2)*(Rabsolu (An n))``.
 Intro; Cut (n:nat)``/(Wn n)<=2*/(Rabsolu (An n))``.
 Intro; Cut (n:nat)``(Wn (S n))/(Wn n)<=3*(Rabsolu (An (S n))/(An n))``.
@@ -354,12 +126,12 @@ Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_
 Apply Rle_lt_trans with ``3*(Rabsolu ((An (S n))/(An n)))``.
 Apply H6.
 Apply Rlt_monotony_contra with ``/3``.
-Apply Rlt_Rinv; Apply Rgt_3_0.
+Apply Rlt_Rinv; Sup0.
 Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]; Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps); Unfold Rdiv in H11; Exact H11.
 Left; Change ``0<(Wn (S n))/(Wn n)``; Unfold Rdiv; Apply Rmult_lt_pos.
 Apply H2.
 Apply Rlt_Rinv; Apply H2.
-Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rgt_3_0].
+Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0].
 Intro; Unfold Rdiv; Rewrite Rabsolu_mult; Rewrite <- Rmult_assoc; Replace ``3`` with ``2*(3*/2)``; [Idtac | Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m; DiscrR]; Apply Rle_trans with ``(Wn (S n))*2*/(Rabsolu (An n))``.
 Rewrite Rmult_assoc; Apply Rle_monotony.
 Left; Apply H2.
@@ -367,7 +139,7 @@ Apply H5.
 Rewrite Rabsolu_Rinv.
 Replace ``(Wn (S n))*2*/(Rabsolu (An n))`` with ``(2*/(Rabsolu (An n)))*(Wn (S n))``; [Idtac | Ring]; Replace ``2*(3*/2)*(Rabsolu (An (S n)))*/(Rabsolu (An n))`` with ``(2*/(Rabsolu (An n)))*((3*/2)*(Rabsolu (An (S n))))``; [Idtac | Ring]; Apply Rle_monotony.
 Left; Apply Rmult_lt_pos.
-Apply Rgt_2_0.
+Sup0.
 Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Apply H.
 Elim (H4 (S n)); Intros; Assumption.
 Apply H.
@@ -378,7 +150,7 @@ Apply Rle_monotony_contra with (Rabsolu (An n)).
 Apply Rabsolu_pos_lt; Apply H.
 Rewrite Rmult_1r; Replace ``(Rabsolu (An n))*((Wn n)*(2*/(Rabsolu (An n))))`` with ``2*(Wn n)*((Rabsolu (An n))*/(Rabsolu (An n)))``; [Idtac | Ring]; Rewrite <- Rinv_r_sym.
 Rewrite Rmult_1r; Apply Rle_monotony_contra with ``/2``.
-Apply Rlt_Rinv; Apply Rgt_2_0.
+Apply Rlt_Rinv; Sup0.
 Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym.
 Rewrite Rmult_1l; Elim (H4 n); Intros; Assumption.
 DiscrR.
@@ -386,12 +158,12 @@ Apply Rabsolu_no_R0; Apply H.
 Red; Intro; Assert H6 := (H2 n); Rewrite H5 in H6; Elim (Rlt_antirefl ? H6).
 Intro; Split.
 Unfold Wn; Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Apply Rle_monotony.
-Left; Apply Rlt_Rinv; Apply Rgt_2_0.
+Left; Apply Rlt_Rinv; Sup0.
 Pattern 1 (Rabsolu (An n)); Rewrite <- Rplus_Or; Rewrite double; Unfold Rminus; Rewrite Rplus_assoc; Apply Rle_compatibility.
 Apply Rle_anti_compatibility with (An n).
 Rewrite Rplus_Or; Rewrite (Rplus_sym (An n)); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply Rle_Rabsolu.
 Unfold Wn; Unfold Rdiv; Repeat Rewrite <- (Rmult_sym ``/2``); Repeat Rewrite Rmult_assoc; Apply Rle_monotony.
-Left; Apply Rlt_Rinv; Apply Rgt_2_0.
+Left; Apply Rlt_Rinv; Sup0.
 Unfold Rminus; Rewrite double; Replace ``3*(Rabsolu (An n))`` with ``(Rabsolu (An n))+(Rabsolu (An n))+(Rabsolu (An n))``; [Idtac | Ring]; Repeat Rewrite Rplus_assoc; Repeat Apply Rle_compatibility.
 Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu.
 Cut (n:nat)``/2*(Rabsolu (An n))<=(Vn n)<=(3*/2)*(Rabsolu (An n))``.
@@ -405,12 +177,12 @@ Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_
 Apply Rle_lt_trans with ``3*(Rabsolu ((An (S n))/(An n)))``.
 Apply H5.
 Apply Rlt_monotony_contra with ``/3``.
-Apply Rlt_Rinv; Apply Rgt_3_0.
+Apply Rlt_Rinv; Sup0.
 Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]; Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps); Unfold Rdiv in H10; Exact H10.
 Left; Change ``0<(Vn (S n))/(Vn n)``; Unfold Rdiv; Apply Rmult_lt_pos.
 Apply H1.
 Apply Rlt_Rinv; Apply H1.
-Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rgt_3_0].
+Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0].
 Intro; Unfold Rdiv; Rewrite Rabsolu_mult; Rewrite <- Rmult_assoc; Replace ``3`` with ``2*(3*/2)``; [Idtac | Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m; DiscrR]; Apply Rle_trans with ``(Vn (S n))*2*/(Rabsolu (An n))``.
 Rewrite Rmult_assoc; Apply Rle_monotony.
 Left; Apply H1.
@@ -418,7 +190,7 @@ Apply H4.
 Rewrite Rabsolu_Rinv.
 Replace ``(Vn (S n))*2*/(Rabsolu (An n))`` with ``(2*/(Rabsolu (An n)))*(Vn (S n))``; [Idtac | Ring]; Replace ``2*(3*/2)*(Rabsolu (An (S n)))*/(Rabsolu (An n))`` with ``(2*/(Rabsolu (An n)))*((3*/2)*(Rabsolu (An (S n))))``; [Idtac | Ring]; Apply Rle_monotony.
 Left; Apply Rmult_lt_pos.
-Apply Rgt_2_0.
+Sup0.
 Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Apply H.
 Elim (H3 (S n)); Intros; Assumption.
 Apply H.
@@ -429,7 +201,7 @@ Apply Rle_monotony_contra with (Rabsolu (An n)).
 Apply Rabsolu_pos_lt; Apply H.
 Rewrite Rmult_1r; Replace ``(Rabsolu (An n))*((Vn n)*(2*/(Rabsolu (An n))))`` with ``2*(Vn n)*((Rabsolu (An n))*/(Rabsolu (An n)))``; [Idtac | Ring]; Rewrite <- Rinv_r_sym.
 Rewrite Rmult_1r; Apply Rle_monotony_contra with ``/2``.
-Apply Rlt_Rinv; Apply Rgt_2_0.
+Apply Rlt_Rinv; Sup0.
 Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym.
 Rewrite Rmult_1l; Elim (H3 n); Intros; Assumption.
 DiscrR.
@@ -437,19 +209,19 @@ Apply Rabsolu_no_R0; Apply H.
 Red; Intro; Assert H5 := (H1 n); Rewrite H4 in H5; Elim (Rlt_antirefl ? H5).
 Intro; Split.
 Unfold Vn; Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Apply Rle_monotony.
-Left; Apply Rlt_Rinv; Apply Rgt_2_0.
+Left; Apply Rlt_Rinv; Sup0.
 Pattern 1 (Rabsolu (An n)); Rewrite <- Rplus_Or; Rewrite double; Rewrite Rplus_assoc; Apply Rle_compatibility.
 Apply Rle_anti_compatibility with ``-(An n)``; Rewrite Rplus_Or; Rewrite <- (Rplus_sym (An n)); Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu.
 Unfold Vn; Unfold Rdiv; Repeat Rewrite <- (Rmult_sym ``/2``); Repeat Rewrite Rmult_assoc; Apply Rle_monotony.
-Left; Apply Rlt_Rinv; Apply Rgt_2_0.
+Left; Apply Rlt_Rinv; Sup0.
 Unfold Rminus; Rewrite double; Replace ``3*(Rabsolu (An n))`` with ``(Rabsolu (An n))+(Rabsolu (An n))+(Rabsolu (An n))``; [Idtac | Ring]; Repeat Rewrite Rplus_assoc; Repeat Apply Rle_compatibility; Apply Rle_Rabsolu.
 Intro; Unfold Wn; Unfold Rdiv; Rewrite <- (Rmult_Or ``/2``); Rewrite <- (Rmult_sym ``/2``); Apply Rlt_monotony.
-Apply Rlt_Rinv; Apply Rgt_2_0.
+Apply Rlt_Rinv; Sup0.
 Apply Rlt_anti_compatibility with (An n); Rewrite Rplus_Or; Unfold Rminus; Rewrite (Rplus_sym (An n)); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply Rle_lt_trans with (Rabsolu (An n)).
 Apply Rle_Rabsolu.
 Rewrite double; Pattern 1 (Rabsolu (An n)); Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rabsolu_pos_lt; Apply H.
 Intro; Unfold Vn; Unfold Rdiv; Rewrite <- (Rmult_Or ``/2``); Rewrite <- (Rmult_sym ``/2``); Apply Rlt_monotony.
-Apply Rlt_Rinv; Apply Rgt_2_0.
+Apply Rlt_Rinv; Sup0.
 Apply Rlt_anti_compatibility with ``-(An n)``; Rewrite Rplus_Or; Unfold Rminus; Rewrite (Rplus_sym ``-(An n)``); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Apply Rle_lt_trans with (Rabsolu (An n)).
 Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu.
 Rewrite double; Pattern 1 (Rabsolu (An n)); Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rabsolu_pos_lt; Apply H.
@@ -598,3 +370,179 @@ Exists (sum_f_R0 An O); Unfold EUn; Exists O; Reflexivity.
 Intro; Elim X; Intros.
 Apply Specif.existT with x; Apply tech10; [Unfold Un_growing; Intro; Rewrite tech5; Pattern 1 (sum_f_R0 An n); Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Apply H | Apply p].
 Qed.
+
+Lemma Alembert_strong_general : (An:nat->R;k:R) ``0<=k<1`` -> ((n:nat)``(An n)<>0``) -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) k) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)).
+Intros.
+Cut (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)).
+Intro Hyp0; Apply Hyp0.
+Apply cv_cauchy_2.
+Apply cauchy_abs.
+Apply cv_cauchy_1.
+Cut (SigT R [l:R](Un_cv [N:nat](sum_f_R0 [i:nat](Rabsolu (An i)) N) l)) -> (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 [i:nat](Rabsolu (An i)) N) l)).
+Intro Hyp; Apply Hyp.
+Apply (Alembert_strong_positive [i:nat](Rabsolu (An i)) k).
+Assumption.
+Intro; Apply Rabsolu_pos_lt; Apply H0.
+Unfold Un_cv.
+Unfold Un_cv in H1.
+Unfold Rdiv.
+Intros.
+Elim (H1 eps H2); Intros.
+Exists x; Intros.
+Rewrite <- Rabsolu_Rinv.
+Rewrite <- Rabsolu_mult.
+Rewrite Rabsolu_Rabsolu.
+Unfold Rdiv in H3; Apply H3; Assumption.
+Apply H0.
+Intro.
+Elim X; Intros.
+Apply existTT with x.
+Assumption.
+Intro.
+Elim X; Intros.
+Apply Specif.existT with x.
+Assumption.
+Qed.
+
+(* Convergence de la SE dans le disque D(O,1/k) *)
+(* le cas k=0 est decrit par le theoreme "Alembert" *)
+Lemma Alembert_strong : (An:nat->R;x,k:R) ``0<k`` -> ((n:nat)``(An n)<>0``) -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) k) -> ``(Rabsolu x)</k`` -> (SigT R [l:R](Pser An x l)).
+Intros.
+Cut (SigT R [l:R](Un_cv [N:nat](sum_f_R0 [i:nat]``(An i)*(pow x i)`` N) l)).
+Intro.
+Elim X; Intros.
+Apply Specif.existT with x0.
+Apply tech12; Assumption.
+Case (total_order_T x R0); Intro.
+Elim s; Intro.
+EApply Alembert_strong_general with ``k*(Rabsolu x)``.
+Split.
+Unfold Rdiv; Apply Rmult_le_pos.
+Left; Assumption.
+Left; Apply Rabsolu_pos_lt.
+Red; Intro; Rewrite H3 in a; Elim (Rlt_antirefl ? a).
+Apply Rlt_monotony_contra with ``/k``.
+Apply Rlt_Rinv; Assumption.
+Rewrite <- Rmult_assoc.
+Rewrite <- Rinv_l_sym.
+Rewrite Rmult_1l.
+Rewrite Rmult_1r; Assumption.
+Red; Intro; Rewrite H3 in H; Elim (Rlt_antirefl ? H).
+Intro; Apply prod_neq_R0.
+Apply H0.
+Apply pow_nonzero.
+Red; Intro; Rewrite H3 in a; Elim (Rlt_antirefl ? a).
+Unfold Un_cv; Unfold Un_cv in H1.
+Intros.
+Cut ``0<eps/(Rabsolu x)``.
+Intro.
+Elim (H1 ``eps/(Rabsolu x)`` H4); Intros.
+Exists x0.
+Intros.
+Replace ``((An (S n))*(pow x (S n)))/((An n)*(pow x n))`` with ``(An (S n))/(An n)*x``.
+Unfold R_dist.
+Rewrite Rabsolu_mult.
+Replace ``(Rabsolu ((An (S n))/(An n)))*(Rabsolu x)-k*(Rabsolu x)`` with ``(Rabsolu x)*((Rabsolu ((An (S n))/(An n)))-k)``; [Idtac | Ring].
+Rewrite Rabsolu_mult.
+Rewrite Rabsolu_Rabsolu.
+Apply Rlt_monotony_contra with ``/(Rabsolu x)``.
+Apply Rlt_Rinv; Apply Rabsolu_pos_lt.
+Red; Intro; Rewrite H7 in a; Elim (Rlt_antirefl ? a).
+Rewrite <- Rmult_assoc.
+Rewrite <- Rinv_l_sym.
+Rewrite Rmult_1l.
+Rewrite <- (Rmult_sym eps).
+Unfold R_dist in H5.
+Unfold Rdiv; Unfold Rdiv in H5; Apply H5; Assumption.
+Apply Rabsolu_no_R0.
+Red; Intro; Rewrite H7 in a; Elim (Rlt_antirefl ? a).
+Unfold Rdiv; Replace (S n) with (plus n (1)); [Idtac | Ring].
+Rewrite pow_add.
+Simpl.
+Rewrite Rmult_1r.
+Rewrite Rinv_Rmult.
+Replace ``(An (plus n (S O)))*((pow x n)*x)*(/(An n)*/(pow x n))`` with ``(An (plus n (S O)))*/(An n)*x*((pow x n)*/(pow x n))``; [Idtac | Ring].
+Rewrite <- Rinv_r_sym.
+Rewrite Rmult_1r; Reflexivity.
+Apply pow_nonzero.
+Red; Intro; Rewrite H7 in a; Elim (Rlt_antirefl ? a).
+Apply H0.
+Apply pow_nonzero.
+Red; Intro; Rewrite H7 in a; Elim (Rlt_antirefl ? a).
+Unfold Rdiv; Apply Rmult_lt_pos.
+Assumption.
+Apply Rlt_Rinv; Apply Rabsolu_pos_lt.
+Red; Intro H7; Rewrite H7 in a; Elim (Rlt_antirefl ? a).
+Apply Specif.existT with (An O).
+Unfold Un_cv.
+Intros.
+Exists O.
+Intros.
+Unfold R_dist.
+Replace (sum_f_R0 [i:nat]``(An i)*(pow x i)`` n) with (An O).
+Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption.
+Induction n.
+Simpl; Ring.
+Rewrite tech5.
+Rewrite <- Hrecn.
+Rewrite b; Simpl; Ring.
+Unfold ge; Apply le_O_n.
+EApply Alembert_strong_general with ``k*(Rabsolu x)``.
+Split.
+Unfold Rdiv; Apply Rmult_le_pos.
+Left; Assumption.
+Left; Apply Rabsolu_pos_lt.
+Red; Intro; Rewrite H3 in r; Elim (Rlt_antirefl ? r).
+Apply Rlt_monotony_contra with ``/k``.
+Apply Rlt_Rinv; Assumption.
+Rewrite <- Rmult_assoc.
+Rewrite <- Rinv_l_sym.
+Rewrite Rmult_1l.
+Rewrite Rmult_1r; Assumption.
+Red; Intro; Rewrite H3 in H; Elim (Rlt_antirefl ? H).
+Intro; Apply prod_neq_R0.
+Apply H0.
+Apply pow_nonzero.
+Red; Intro; Rewrite H3 in r; Elim (Rlt_antirefl ? r).
+Unfold Un_cv; Unfold Un_cv in H1.
+Intros.
+Cut ``0<eps/(Rabsolu x)``.
+Intro.
+Elim (H1 ``eps/(Rabsolu x)`` H4); Intros.
+Exists x0.
+Intros.
+Replace ``((An (S n))*(pow x (S n)))/((An n)*(pow x n))`` with ``(An (S n))/(An n)*x``.
+Unfold R_dist.
+Rewrite Rabsolu_mult.
+Replace ``(Rabsolu ((An (S n))/(An n)))*(Rabsolu x)-k*(Rabsolu x)`` with ``(Rabsolu x)*((Rabsolu ((An (S n))/(An n)))-k)``; [Idtac | Ring].
+Rewrite Rabsolu_mult.
+Rewrite Rabsolu_Rabsolu.
+Apply Rlt_monotony_contra with ``/(Rabsolu x)``.
+Apply Rlt_Rinv; Apply Rabsolu_pos_lt.
+Red; Intro; Rewrite H7 in r; Elim (Rlt_antirefl ? r).
+Rewrite <- Rmult_assoc.
+Rewrite <- Rinv_l_sym.
+Rewrite Rmult_1l.
+Rewrite <- (Rmult_sym eps).
+Unfold R_dist in H5.
+Unfold Rdiv; Unfold Rdiv in H5; Apply H5; Assumption.
+Apply Rabsolu_no_R0.
+Red; Intro; Rewrite H7 in r; Elim (Rlt_antirefl ? r).
+Unfold Rdiv; Replace (S n) with (plus n (1)); [Idtac | Ring].
+Rewrite pow_add.
+Simpl.
+Rewrite Rmult_1r.
+Rewrite Rinv_Rmult.
+Replace ``(An (plus n (S O)))*((pow x n)*x)*(/(An n)*/(pow x n))`` with ``(An (plus n (S O)))*/(An n)*x*((pow x n)*/(pow x n))``; [Idtac | Ring].
+Rewrite <- Rinv_r_sym.
+Rewrite Rmult_1r; Reflexivity.
+Apply pow_nonzero.
+Red; Intro; Rewrite H7 in r; Elim (Rlt_antirefl ? r).
+Apply H0.
+Apply pow_nonzero.
+Red; Intro; Rewrite H7 in r; Elim (Rlt_antirefl ? r).
+Unfold Rdiv; Apply Rmult_lt_pos.
+Assumption.
+Apply Rlt_Rinv; Apply Rabsolu_pos_lt.
+Red; Intro H7; Rewrite H7 in r; Elim (Rlt_antirefl ? r).
+Qed.
\ No newline at end of file